Which Came First the Chicken or the Egg? Inverse Functions

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Objective

SWBAT find and create inverse functions, explaining why some inverse functions do not exist. SWBAT clarify their explanation using details.

Big Idea

Students construct their own understanding of inverse functions through an exploratory function sort!

Entry Ticket: Planning a School Trip

15 minutes

As usual, I start this lesson with an Entry Ticket.  Today's opener targets MP4 and MP3. Modeling is promoted by using a real-life scenario or context. MP3 is accessed by the entry ticket usually through the use of a Turn and Talk where students have an opportunity to engage in a meaningful conversation with a classmate. I give students about 10 minutes to work on the Entry Ticket, 2-3 minutes to discuss with a peer, and an additional few minutes to have a class-wide discussion to flush out the main points of the ticket. 

For this lesson, I hook students through the context of planning a trip to New York City (Entry Ticket: School Trip as an Inverse Function). I focus on the common core standard A-CED4  by asking students to rearrange a formula (D=R*T), so it can be more easily solved for the variable of interest (T = Time). I then have students complete the table for various average speeds for the trip to New York and ask student to graph the table of values along with a graph of the linear function. 

The underlying concept of the lesson is revealed when students engage in swapping the axes and generating a second linear function. Th idea is to get students to think about what it means to swap axes and what effect it might have on the function. I intentionally have students complete this problem before going through the steps to generate inverse functions because I feel it provides an opportunity for students to construct their own knowledge (for more see Constructivist point of view). 

Think, Pair, Share: Class Trip as an Inverse Function

20 minutes

Next I will ask my students to complete a Think-Pair-Share on the School Trip exercise from the opening of the lesson. I plan to give students about 10 minutes to compare and contrast answers using the prompts at the end of the Entry Ticket for this lesson. I pair students purposefully, so that they have a more genuine chance to engage in meaningful Academic Conversations. Giving students time to process and work on the information on their own helps them create their own arguments. Giving time to talk to a peer provides the chance to critique the reasoning of others (MP3). 

After about eight minutes, I will give students a two-minute warning and then hold a class discussion of the problem. I will ask each pair of students to share one interesting point about their conversation, emphasizing the "why" of its importance. Again, academic conversations are being stressed, with the specific skill of Elaborate and Clarify being highlighted in the group discussion. I am not only asking students to summarize their conversations, but also want them to provide supporting and relevant details to further their argument/point in a class-wide setting. 

 

Class Practice: Chickens and Eggs as Inverse Functions

20 minutes

Today's Focus Lesson is brief and students take the lead in doing mopst of the work. The Power Point Slides: Which Came First...Inverse Functions introduce some ideas related to inverse functions. I want students to focus on the concept of inverse functions, so I use a linear function in both examples. In both examples, my students will complete a table of values, then Turn-and-Talk with a partner to identify the function that best describes the relationship between the variables.  

Afterward, I plan to lead the class through a series of steps to create an inverse function from the original function, in this case f(C) = C + 2.  I cue students to be taking Two-Column Notes during this exercise so they have a reference they can refer to when solving similar problems. I model the two column note format on the power point slides and write in the steps as the class identifies them in solving the problem. 

 

Think, Pair, Share: Which Came First?

20 minutes

At this point in the lesson, I ask students to complete a Think, Pair, Share about the meaning of an inverse function in the context of a real world problem. I want students to focus on the meaning of inverse functions in a way that is broader and more applicable than simply solving for x and exchanging the x and y variables. The context of chickens and eggs works because it is a hook that students are familiar with, and, the core of a well known question about order of origin, Which_came_first the chicken or the Egg?

 

Exit Ticket and Homework: Create Your Own Inverse Function

15 minutes

For the Exit Ticket and Homework: Create Your Own Inverse Function students complete an Idea Organizer that helps them organize their thoughts and puts an emphasis on writing in the classroom. 

For this particular Idea Organizer, students have to identify a function that describes a relationship in a context. If students are having difficulty, I would get them started by asking for examples and jotting a few on the board. I have students generate their own functions because it provides students with additional practice at previously taught concepts in this unit (function notation in particular). 

Students are also asked to calculate and interpret the inverse of the function they identify. This task gets at the heart of the day's lesson as students not only need to look for and see regularity in repeated reasoning, MP.8, but they also have to interpret the meaning of an inverse function. I firmly believe that I have not done by job as an educator if students cannot use the skills and concepts outside of the classroom walls, and this exercise is one small example of helping students to bridge that gap that too often occurs between the classroom and the real world.